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  "body": "\n## Summary\n\nThe mod $2$ Steenrod algebra $\\mathcal{A}_2$ is the graded algebra over $\\Z/2\\Z$ generated by the [[Steenrod squares|VR7RHP41M0]], subject to the [[Adem relations|GS8SCGYPX7]].\nIt is the algebra of all stable [[cohomology operations|K8T15QHN3W]] on mod $2$ cohomology.\nSo the mod $2$ cohomology of every space is a module over it.\nFor each odd prime $p$ there is an analogous algebra built from the reduced powers and the Bockstein.\nThe Steenrod algebra is a Hopf algebra, and it is the input of the [[Adams spectral sequence|DD4D0EKX74]], which computes stable homotopy groups.\n\n## Explanations\n\n### L1 human\n\nSteenrod squares can be composed, and compositions of them are again cohomology operations.\nThe Steenrod algebra collects all such compositions and their sums.\nThe squares do not commute freely: certain compositions can be rewritten in terms of others, and the Adem relations list the rules.\nFor example, applying $\\operatorname{Sq}^1$ twice always gives zero.\n",
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